Lemma 101.21.5. Let \mathcal{X} be an algebraic stack. Consider a cartesian diagram
\xymatrix{ U \ar[d] & F \ar[l]^ p \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l] }
where U is an algebraic space, k is a field, and U \to \mathcal{X} is flat and locally of finite presentation. Let z \in |F| be such that F \to \mathop{\mathrm{Spec}}(k) is unramified at z. Then, after replacing U by an open subspace containing p(z), the morphism
U \longrightarrow \mathcal{X}
is étale.
Proof.
Since f : U \to \mathcal{X} is flat and locally of finite presentation there exists a maximal open W(f) \subset U such that the restriction f|_{W(f)} : W(f) \to \mathcal{X} is étale, see Properties of Stacks, Remark 100.9.20 (5). Hence all we need to do is prove that p(z) is a point of W(f). Moreover, the remark referenced above also shows the formation of W(f) commutes with arbitrary base change by a morphism which is representable by algebraic spaces. Hence it suffices to show that the morphism F \to \mathop{\mathrm{Spec}}(k) is étale at z. Since it is flat and locally of finite presentation as a base change of U \to \mathcal{X} and since F \to \mathop{\mathrm{Spec}}(k) is unramified at z by assumption, this follows from Morphisms of Spaces, Lemma 67.39.12.
\square
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